1. Field of the Invention
The present invention is directed towards optical systems. In particular the present invention is directed towards a method of designing an optical system consisting of two reflecting surfaces and other optical systems consisting of multiple surface, reflecting or refracting designed by the method.
2. Description of the Related Technology
Designing optical systems that use free-form surfaces is challenging, even when designing a system that uses only a single surface. Free-form surfaces are natural choices for optical systems that lack symmetry. By “free-form” it is meant that the surface is not required to conform to a fixed class of shapes, i.e. it need not be rotationally symmetric and may have a plurality of non-symmetrical undulations. An early example of a free-form surface is a progressive lens (C. W. Kanolt. Multifocal Opthalmic Lenses. U.S. Pat. No. 2,878,721, Mar. 24, 1959.), such as those used in corrective lenses. Another example that uses free-form surfaces is the Polaroid SX-70 camera. Until recently, the grinding of free-form surfaces was extremely difficult, but now with technology developed by the DARPA conformal optics program, techniques, such as raster grinding are commercially available.
Free-form surfaces can play a role in numerous applications that by their nature lack rotational symmetry, but methods for the design of free-form surfaces are in their infancy. Illumination is a natural application area. However, for illumination, even the problem of controlling the intensity from a single point source with a single reflector is quite challenging.
The design of multiple free-form surface systems is considerably more complicated than that of single surface free-form systems. A coupled pair of free-form lenses has been considered in shaping laser beams. Two reflector systems for illumination have also been investigated. Examples include work by Oliker “On design of free-form refractive beam shapers, sensitivity to figure error, and convexity of lenses”, Vol. 25, No. 12, December 2008, J. Opt. Soc. Am. A.
Additionally, the problem of finding a mirror that induced a prescribed projection has been considered. In solving this problem it is assumed that a pinhole camera views a mirror. This case is unusual, in that the pinhole is considered as a single source from which rays emanate via the reversibility of geometric optics. In solving this problem, a surface S is to be imaged onto a plane I in a prescribed way. A mapping of T: I→S is part of the data of the problem. This is shown in FIG. 1, where given a correspondence, T, that points from an image plane to points on a target surface, one can attempt to define a vector field W that is normal to a mirror surface that realizes the correspondence. The mapping corresponds to controlling the distortion of the mirror. Given such a correspondence, a vector field W is then defined on some subset of R3 via the construction given in FIG. 1. If a solution M exists then W will be perpendicular to it.
The length of W is irrelevant, it is only important that each tangent plane TpM of M be orthogonal to W. This gives rise to a planar distribution, i.e. an assignment of a plane, Ep, to each point p in the domain of interest, in which M is an integral surface of that distribution. That is to say the tangent planes of M coincide with the planes of the distribution.
It is unlikely, but if ∇×W=0 then W=∇φ for some φ and the level surfaces φ (x,y, z)=C are each a solution to the problem. In cases such as this, where an open set is foliated by solution surfaces, the distribution is said to integrable. It may be that W is not a gradient, but a multiple of a gradient, which occurs if (∇×W)·W=0. This is the vector calculus version of a statement in terms of differential forms: if θ is the 1-form dual to W, then the distribution is integrable in an open set if θdθ=0 in that set.
While some work with free-form surfaces has been pursued, there has not been a system and method that has been able to accurately design a two mirror system that is able to use free-form surfaces.